Sometimes functions are defined like `f(x)=max{sinx,cosx}`, then `f(x)` is splitted like `f(x)={{:(cosx, x in (0,(pi)/(4)]),(sinx, x in ((pi)/(4),(pi)/(2)]):}` etc.
If `f(x)=max{x^(2),2^(x)}`,then if `x in (0,1)`, `f(x)=`

Options

A. `x^(2)`
B. `2^(x)`
C. both are equal
D. none of these

Answer

To solve the problem where \( f(x) = \max\{x^2, 2^x\} \) for \( x \in (0, 1) \), we will analyze the two functions \( x^2 \) and \( 2^x \) over the interval \( (0, 1) \) to determine which function is greater in that range.

### Step 1: Evaluate the functions at the endpoints of the interval
First, we will evaluate \( f(x) \) at the endpoints of the interval \( x = 0 \) and \( x = 1 \).

- For \( x = 0 \):
\[ f(0) = \max\{0^2, 2^0\} = \max\{0, 1\} = 1 \]

- For \( x = 1 \):
\[ f(1) = \max\{1^2, 2^1\} = \max\{1, 2\} = 2 \]

### Step 2: Analyze the behavior of the functions in the interval \( (0, 1) \)
Next, we need to check the behavior of both functions \( x^2 \) and \( 2^x \) within the interval \( (0, 1) \).

- The function \( x^2 \) is a parabola that opens upwards and is increasing in the interval \( (0, 1) \).
- The function \( 2^x \) is an exponential function that is also increasing in the interval \( (0, 1) \).

### Step 3: Find the point where the two functions intersect
To find where \( x^2 = 2^x \), we can set up the equation:
\[ x^2 = 2^x \]
This equation is not straightforward to solve algebraically, so we can analyze it graphically or numerically.

### Step 4: Check values in the interval
We can check some values in the interval \( (0, 1) \) to see which function is greater.

- For \( x = 0.5 \):
\[ f(0.5) = \max\{(0.5)^2, 2^{0.5}\} = \max\{0.25, \sqrt{2}\} \approx \max\{0.25, 1.414\} = 1.414 \]

- For \( x = 0.1 \):
\[ f(0.1) = \max\{(0.1)^2, 2^{0.1}\} = \max\{0.01, 1.0718\} \approx 1.0718 \]

From these evaluations, we can see that \( 2^x \) is greater than \( x^2 \) for values in \( (0, 1) \).

### Step 5: Conclusion
Since \( 2^x \) is greater than \( x^2 \) for all \( x \in (0, 1) \), we conclude:
\[ f(x) = 2^x \text{ for } x \in (0, 1) \]

### Final Answer
Thus, the final answer is:
\[ f(x) = 2^x \text{ for } x \in (0, 1) \]